Painleve方程式の特殊解 (Painleve系と超幾何系)

(1)

Painlev\’e 方程式の特殊解

大阪大学理学研究科・大山陽介 (Yousuke Ohyama)

School ofScience, Osaka University

1 Introduction

In this paper

we

will study the third Painlev\’e equations $P_{111}(\alpha,\beta,\gamma, \delta)$

$y’= \frac{1}{y}y^{\prime 2}-\frac{y’}{x}+\frac{\alpha y^{2}+\beta}{x}+\gamma y^{3}+\frac{\delta}{y}$, (1)

for $\gamma=0$ and $a\mathit{6}\neq 0$

.

The values of complex parameters $\alpha$,$\beta,\gamma,\delta$ ofthe third Painlev\’e equations

satisfy

one

of four

cases:

(Q)

a

$=0,\gamma=0$ (or $\beta=0,\delta=0$),

$(D_{8})\gamma=0$, $\delta=0\alpha\beta\neq 0$

$(D_{7})\gamma=0$, $\alpha\delta\neq 0$ (or $\delta$ _{$=0,\beta\gamma\neq 0$}),

$(D_{6})\gamma\delta\neq 0$.

In the

case

(Q),$\mathrm{f}1_{\mathrm{I}1}$

are

sovable by quadraturs ([11], [23]).

Since

allof solutions

of(Q)

are

classicalinUmemura’s meaning ([27]),

we

do not include the equation

(Q) in the Painleve equations. Gromak ([4]) also excluded the

case

(Q). The

type $D_{6}$, which is generic case, is studied in many articles ([23], [18], [28]). The

equations of type $D_{7}$ and $D_{8}$

are

missed in most study of the third Painleve

equation

so

far. Gromak studied the type $D_{7}$ in [2].

Recently Sakai pointed out the significance ofthe type $D_{7}$ and $D_{8}$ in [24].

He showed that the spaces of the initial values for the type Dq, $D_{7}$ and $D_{8}$

are

different from each other. The vertical leaves become

asum

of rational

curves, whose intersection diagrams

are

the root lattices $D_{6}$, $D_{7}$ and $D_{8}$ in

each

case.

This is the origin ofthe

name

of each type. Moreover the Backlund

transformation

groups

of the equation oftype Dq, $D_{7}$ and $D_{8}$

are

$W(A_{1}\oplus A_{1})$,

$W(A_{1})$ and $\mathrm{Z}_{2}$, respectively. Prom Sakai’s viewpoint,

we

should study eight

(not six) type of Painlev\’e equations.

This papar is asupplement of Okamoto’s series of four papers “Studies

on

the Painlev\’e equations” ([20], [21], [22], [23]) published in $1980\mathrm{s}$

.

We will

study Hamitonian structures, atransformation

group,

$\tau$-functions and special

solutions oftype $D_{7}$

.

The type$D_{8}$ reduced to special

case

of$D_{6}$ by aquadratic

transformation. We will comment about the type $D_{8}$ in Theorem

3.1.

We will study the equation $P_{111}’(\alpha,\beta,\gamma, \delta)$

$q’= \frac{q^{\prime 2}}{q}-\frac{q’}{t}+\frac{\alpha q^{2}}{4t^{2}}+\frac{\beta}{4t}+\frac{\gamma q^{3}}{4t^{2}}+\frac{\delta}{4q}$ , (2)

数理解析研究所講究録 1239 巻 2001 年 22-40

(2)

which is equivalent to $P_{1\mathrm{I}1}$ by

$t=x^{2}$, _$y=xq$. ₍₃₎

We will consider$P_{111}’$ instead of_{$P_{111}([23])$}, because the action ofatransformaion

group

on

$P_{111}’$ is simplerthan $ffl_{11}$

.

By change of variables

$x=\lambda x_{1}$, _{$y=\mu y_{1}$} (Ap $\neq 0$), (4)

we

can

normalize the parameter $(\alpha, \beta,\gamma, \delta)$. Essentially, the equations of type

$D_{6}$ have two comlex parameters, the equations of type _{$D_{7}$} have

one

comlex

parameter _{and the equations of type} $D_{8}$ have

no

comlex parameters. For the

type $D_{7}$,

we can

take standard form

$q’= \frac{q^{\prime 2}}{q}-\frac{q’}{t}-\frac{q^{2}}{t^{2}}+\frac{a}{t}-\frac{1}{q}$

.

(5)

This equation is the main object in this paper.

An algebraic solution of the third Painleve equations

are

found by

Lukashe-vich ([11], [12]). Gromak classified all algebraic solutions of the third Painleve

equations not only for type $D_{6}$ but also for type _{$D_{7}([3], [4])$}. If_$a$ is an integer,

(5) has

one

and only

one

algebraic solution. If $a$ is not

an

integer, (5) has

no

rational solution. The equation oftype $D_{8}$ has two rational solutions.

The algebraic solutions ofPainleve equations

are

studied by many authors.

There

are

many works by Belorussian school (see [5]). After Okamoto showed

that the transformation groups of Painleve equations are isomorphic to affine

Wyle groups, it iseasyto understand their works. Murata gives classificationof

algebraic solutions of the second, third and fourth Painleve equations in terms

of affine Wyle groups ([17], [18]). Kitaev-Law-McLeod [9] classified rational

solutions of the fifth Painleve equations. Mazzocco [15] classified rational

solu-tions of the sixth Painleve equations. Any algebraic solutions become rational

for the fifth Painleve equations (announcedin [30]). Algebraic solutions for the

sixth Painleve equations

are

very interesiting ([6], [7], [1], [14], [26], [19], [10])

and they

are

not classified yet. From second to fifth Painleve equations, all

algebraic solutions turn to be rational except for type $D_{7}$.

All of algebraic solutions of (5)

are

transformed to each other by the affine

Wyle group $W(A_{1})$

.

But it is difficult to calculate all algebraic solutions by the

direct action of the affine Wyle group. If

we

consider $\tau$-functions, the action

becomes very simple. The action of the affine Wyle group reduces to the Toda

equation

on

$\tau$-functions. For the second Painleve equations, the

Yablonskii-Vorob’ev polynomials give transformations of $\tau$-functions([29], [22]). There

are

similar polynomials for other Painleve equations ([22] for Painleve $\mathrm{I}\mathrm{V}$, [26]

and [19] for type Dq, $\mathrm{V}$ and $\mathrm{V}\mathrm{I}$). Although the solution of the third Painleve

equations of type $D_{7}$ is algebraic, the action of the affine Wyle group is given

by polynomials. This is

an

analogue of Umemura’s polynomials for the sixth

Painleve’ equations ([26]).

Yablonskii-Vorob’ev polynomials

or

Umemura’s polynomials

are

related to

Shur polynomials ([8], [25]). It is

an

open problem to represent

our

new

poly-nomials by Shur polynomials.

In section four,

we

will study transcendental classical functions of the

equa-tionsoftype$D_{7}$

.

Thethird Painleve equationoftype$D_{7}$ have $0$ like the second

(3)

Painleve equation. The second Painleve equation has also

one

parameter and

the B\"a&lundtransformation

group

istheaffineWyle

group

$W(A_{1})$

.

Thesecond

Painleve equation has transcendental classical functions which

are

reduced to

Airy functions ([27]).

On

the contrary, the third Painlev\’e equation oftype $D_{7}$

does not have transcendental classical functions, followingtheidea of Umemura

and Watanabe. The third Painlev6 equation of type $D_{6}$ has transcendental

classical functions which

are

reduced to Bessel functions ([28]).

The author would give thanks to Prof. H. Watanabe and Prof. H. Sakai for

fruitful discussions. Thiswork is supported inpartby Japan Societyfor the

PrO-motion of

Science

under Grand-in Aid for Scientific Research (No. 12640174).

2Third

Painleve

equation

In this section

we

will review basic facts

on

the third Painlev6 equations ([23]).

We will write (1)

as

$ffl_{11}(\alpha,\beta,\gamma,\delta)$ and (2)

as

$P_{111}’(\alpha,\beta, \gamma,\delta)$

.

2.1 Fundamental transform

Although the third Painlev6equationshave fourcomplex parameters, essensially

they have two complex parameters by simple transformation.

Theorem 2.1. (1) By the change

_of

variables

$t=x^{2}$, _$y=xq$, (6)

$ffl_{11}(\alpha,\beta,\gamma,\delta)$ and $P_{111}’(\alpha, \beta,\gamma,\delta)$

are

equivaliant.

ByTheorem 2.1 (3), thethird Painleve equation oftype$D_{8}$ reduced to type

$D_{6}$

.

By Theorem 2.1 (4), the third Painleve equations $P_{111}’$ oftype $D_{7}$

can

be

normalized to (5).

2.2 Hamiltonian system

The Hamiltonian associated with (5) is

$H= \frac{1}{t}(f^{2}g^{2}-a_{1}fg+tg+\frac{1}{2}f)$ , (10)

(4)

(11)

where $f=q$ and $a=1+a_{1}([24])$. The Hamiltonian _system

_%(ai)

_is

$\{_{\frac{\frac{df}{dgdt}}{dt}=\frac{1}{t}(2fg^{2}-a_{1}g+\frac{1}{2})}=\frac{1}{-t}(2f^{2}g-a_{1}f+t),$

.

We take

an

autiliary Hamiltonian

$h(t)=tH+a_{1}^{2}/4$. ₍₁₂₎ Then

we

have $\{$ $f(t)=- \frac{1-2a_{1}h’(t)+2th’(t)}{4h(t)^{2}},$, $g(t)=h’(t)$. (13) Therefore

we

have

Proposition 2.2. $h$

satisfies

the

differential

equation

$(th’(t))^{2}+4h’(t)^{2}$_(th’$(t)-\mathrm{h}(\mathrm{t})$) _{$+a_{1}h’(t)- \frac{1}{4}=0$}. (14)

Inversely, for asolution of$h(t)$ of (14),

we

have asolution $(f,g)$ of (11) by

(13) if

$\frac{d^{2}h}{dt^{2}}\neq 0$.

Proposition 2.3. TAere exists the one-tO-One correspondence

_from

a solution

$h$

of

(14) and

a

solution _$(f,g)$

of

(11).

The equation (14) admits asingular solution of the form

$h=\lambda t+\mu$,

$- \frac{1}{4}+a_{1}\lambda-4\lambda^{2}\mu=0$.

2.3 Transformation group

The transformation group of the third Painleve equation of type $D_{7}$ is

isomor-phic to the affine Wyle group $W(A_{1})([24])$. The geneators of$W(A_{1})$

are

given

by

$\pi$ :_{$a_{1}arrow-a_{1}$}, (15)

$s:a_{1}arrow-1-a_{1}$

.

(16)

We will show the explicit expression ofthe action of $W(A_{1})$.

Theorem 2.4. 1)

_If

$(f(t),g(t))$

satisfies

$H(a_{1})$,

$(F, G)=(-f(-t)+ \frac{a_{1}}{g(-t)}-\frac{1}{2g(-t)^{2}},$ $-g(-t))$

satisfies

$H(-a_{1})$

.

(5)

$\mathit{2})If(f(t),g(t))$

satisfies

$H(a_{1})$,

$(F, G)=(-2tg(-t)$ , $\frac{f(-t)}{2t})$

satisfies

$H(-1-a_{1})$

.

By Theorem 2.4,

we

have aB\"acklund transformation which gives

$\pi$$\circ s:a_{1}arrow a_{1}+1$,

as

follows:

$(f,g) arrow(\frac{-2t^{2}}{f(t)^{2}}+\frac{2(1+a_{1})t}{f(t)}-2tg(t)$,$\frac{f(t)}{2t})$

.

(17)

This transformation is found by Gromak ([2]).

2.4 r-function

We define the $\tau$-function of$P_{\mathrm{I}11}’$

.

For any solution (f,g), the $\tau$-function $\tau(t)$ is

defined by

$\frac{d}{dt}\log\tau(t)=H(f,g,$t), (18)

uP to constant multiplication.

Theorem 2.5. The $\tau$

-function

$\mathrm{r}(\mathrm{t})$ is holomorphic in _{$\mathbb{C}-\{0\}$} and has simple

zeros.

In most papers, the third Painle6 equations of type $D_{6}$

are

represented

as

monodromy preserving deformation. Recently Kawamuko and Sakai showed

that theequationsoftype $D_{7}$ and $D_{8}$

are

represented

as

monodromy preserving

deformations. Therefore the holomorphicity of the $\tau$-function is followed from

Miwa’s result ([16], [13]).

Here

we

will give direct proofof the holomorphicityof the $\tau$-function Prom

(14), if the auxiliary Hamiltonian $h$ has apole at _{$t=t_{0}(t_{0}\neq 0)$},

$h \sim\frac{t_{0}}{t-t_{0}}+O((t-t_{0})^{0})$,

where $O((t-t_{0})^{0})$ is the Landau’s $O$

.

Therefore

we

have

$H= \frac{1}{t}(h-\frac{a_{1}^{2}}{4})\sim\frac{1}{t-t_{0}}+O((t-t_{0})^{0})$

.

By the definiion of the $\tau$-function, $\tau(t)$ has asimple

zero

at $t=t_{0}$

.

2.5 Toda equation

Let h$=h(t,$_f,g,$a_{1})$ be

an

auxiliaryHamiltonian (12). Wedifine

anew

auxiliary

function $h_{1}$ by

$h_{1}=h-fg+ \frac{2a_{1}+1}{4}$

.

(19)

(6)

We set $(F, G)$ is the Backlund transformation of $(f,g)$ by $\pi\circ s$:

$(F,G)=( \frac{-2t^{2}}{f^{2}}+\frac{2(1+a_{1})t}{f}-2tg$,$\frac{f}{2t})$ .

As the

same

as

(13),

we

have

$\{$

$f=2th_{1}’$,

$-1+2h_{1}’+2a_{1}h_{1}’+2xh_{1}’$

$g=\overline{8th_{1}^{\prime 2}}$.

(20)

Lemma 2.6. $h_{1}(t, f,g, a_{1})$ equals the auxiliary Hamiltonian $h(t, F, G, a_{1}+1)$.

Proof

We have

$H(t, F,G,a_{1}+1)=H(t, f,g,a_{1})- \frac{fg}{t}$ ₍₂₁₎

by direct calculation. Therefore

$h(t, F, G,a_{1}+1)$ $=$ tH$(t, F,G, a_{1}+1)+ \frac{(a_{1}+1)^{2}}{4}$ $=$ tH$(t, f,g, a_{1})-fg+ \frac{(a_{1}+1)^{2}}{4}$

set

$(f_{n},g_{n})=(\ell^{n}(f),\ell^{n}(g))$,

which is asolution for $a_{1}+n$. Let $\tau_{n}$ be afunction defeined by

$\frac{d}{dt}\tau_{n}=H(t, f_{n}, g_{n}, a_{1}+n)$.

Theorem 2.7. $\tau_{n}$ satisfy the Toda equation

$\frac{d}{dt}t\frac{d}{dt}\log\tau_{n}=c(n)\frac{\tau_{n-1}\tau_{n+1}}{\tau_{n}^{2}}$ , (24)

for

some constants $c(n)$

.

(7)

Proof.

$X_{n-1}-X_{n}= \frac{th_{n}’}{h_{n}},=t\frac{d}{dt}\log h_{n}’$. (26)

Prom (25) and (26),

we

obtain

$h_{n}’=c(n) \frac{\tau_{n-1}\tau_{n+1}}{\tau_{n}^{2}}$

.

$\square$

3Polynomial generated by special solution

3.1 Algebraic solution

Theorem 3.1. The third Painleve equation

_of

type $D_{8}$ does not have

trancen-dental classical solutiotes. The third Painleve equation

_of

type $D_{8}$ has two

ratiO-nal solutions. There

are no

more

algebraic solutions.

The first part

comes

ffom [28] and the second part

comes

from ([18]) by

the transformation in Theorem 2.1 (3). Actually, $ffl_{11}(\alpha,\beta, 0,0)$ has constant

solutions $y=\pm\sqrt{-\beta}/\alpha$

.

Lukashevich found special algebraic solutions and Gromak classified all

al-gebraic solutions for type $D_{7}$:

Theorem 3.2. $([\mathit{1}\mathit{1}J, [12])$ In

case

$a_{1}=-1$, _{$H(a_{1})$} has

an

algebraic solution

$f(t)=-t^{2/3}$_, $g(t)= \frac{1}{6t^{2/3}}-\frac{1}{2t^{1/3}}$

.

$H(a_{1})$ has

one

and only

one

algebraic solution

if

$a_{1}$ is

an

integer. These

alge-braic solutions

are

_transformed

by the the Bdcklund

_{transformation}

$\ell^{n}$.

(8)

3.2 Sequence of algebraic solution

First few algebraic solutions in Thorem 3.2 by the Backlund transformation $\ell$

is as followes. If $a_{1}=0$ $(f,g)=( \frac{-t^{\frac{1}{3}}}{3}-t^{\frac{2}{3}}$,$\frac{-1}{2t^{\frac{1}{3}}})$ . If$a_{1}=1$ $(f,g)=( \frac{-5t^{2}\S-12t-9t^{\frac{4}{3}}}{(1+3t^{1}\S)^{2}}$,$\frac{-1-3t^{\frac{1}{3}}}{6t^{2}\S})$ , If $a_{1}=2$, $(f, g)=( \frac{-35x^{1}\S-315x^{2}\S-990x-1350x^{\frac{4}{3}}-891x^{\frac{5}{3}}-243x^{2}}{3(5+12x^{\frac{1}{3}}+9x^{\frac{2}{3}})^{2}}$, $\frac{-5-12t^{\frac{1}{3}}-9t^{\frac{2}{3}}}{2(1+3t^{\frac{1}{3}})^{2}t^{\frac{1}{3}}})$

We

can

calculate $\tau$-functions of algebraic solutions by the Toda equation

(24)

more

easily. From

now on we

set $\tau_{n}(t)$

as

the $\tau$-function of the algebraic

solution for $a_{1}=n$.

Theorem 3.3. Let $s=3t^{1/3}$. _Then we _have

$\tau_{n}(t)=\exp(-\frac{1}{2}ns-\frac{s^{2}}{8})s^{-d_{\mathfrak{n}}/12}S_{n}(s)$,

up to constant multiplication. Here $d_{n}$ is

$d_{n}=\{$

$9n^{2}-1$ $n$ is even,

$9n^{2}-4$ $n$ is odd.

(27)

$S_{n}(s)$ are monic polynomials

of

$s$ with integral

coefficients.

$S_{n}(0)\neq 0$ and

$nS_{n}(s)^{2}+sS_{n}(s)^{2}-2S_{n}(s)S_{n}’(s)+2sS_{n}’(s)^{2}-2sS_{n}(s)S_{n}’(s)$

$=\{$

$sS_{n+1}(s)S_{n-1}(s)$ $n$ is even, (28)

$S_{n+1}(s)S_{n-1}(s)$ $n$ is odd.

Proof.

We

assume

that

$\tau_{n}(t)=\exp(-\frac{3}{2}t^{1/3}-\frac{9}{8}t^{2/3})t^{-c_{n}/36}T_{n}(s)$.

By the Toda equation (24)

we

have arecurrent relation

$nT_{n}(s)^{2}+sT_{n}(s)^{2}-2T_{n}(s)T_{n}’(s)+2sT_{n}’(s)^{2}-2sT_{n}(s)T_{n}’(s)$ (29) $=T_{n+1}(s)T_{n-1}(s)$, and $2c_{n}+12=c_{n-1}+c_{n+1}$. We set $T_{n}(s)=a_{0}^{(n)}+a_{1}^{(n)}s+a_{2}^{(n)}s^{2}+O(s^{3})$.

29

(9)

Then the left hand side of (29) is

$(na_{0}^{(n)}-2a_{1}^{(n)})a_{0}^{(n)}+((a_{0}^{(n)})^{2}+2na_{0}^{(n)}a_{1}^{(n)}-8a_{0}^{(n)}a_{2}^{(n)})s+O(s^{2})$.

We will

see

that if$n$ is even, $na_{0}^{(n)}-2a_{1}^{(n)}=0$ and $(a_{0}^{(n)})^{2}+2na_{0}^{(n)}a_{1}^{(n)}$

we

have

$na_{0}^{(n)}-2a_{1}^{(n)}=a_{0}^{(n-1)}a_{0}^{(n+1)}$.

Therefore $a_{0}^{(n+1)}$ is also

an

odd integer.

Assume that $n$ is

even

We set

By (14)

$(th’(t))^{2}+4h’(t)^{2}(th’(t)-h(t))+nh’(t)- \frac{1}{4}=\frac{(4-9n^{2}+d_{n})k_{1}^{2}}{81}t^{-4/3}+O(t^{-1})$.

Thus when $n$ is even,

we

have

$k_{1}= \frac{a_{1}^{(n)}}{a_{0}^{(n)}}-\frac{n}{2}=0$.

(10)

By Theorem 2.5, $S_{n}(s)$ have simple

zeros.

The polynomials $S_{n}(s)$

are

ana-logueofYablonskii-Vorob’evpolynomialsfor the second Painleve equations. It is

aconjecture that $S_{n}(s)$

can

be represented by Shur polynomials like

Yablonskii-Vorob’ev polynomials ([8]).

We will list $S_{n}(s)$ for $n=0,1,2,3,4,5$.

$S_{0}(s)=1$, $S_{1}(s)=1$, $S_{2}(s)=s+1$, $S_{3}(s)=s^{2}+4s+5$, $S_{4}(s)=s^{4}+10s^{3}+40s^{2}+70s+35$, $S_{5}(s)=s^{6}+20s^{5}+175s^{4}+840s^{3}+2275s^{2}+3220s+1925$.

4Transcendental classical solutions

4.1 Main

Theorem

(30)

ThethirdPainleve equations have classical solutions writtenbyBessel functions.

The third Painleve’ equations of type $D_{6}$ is in [24].

TheHamiltonian form ofthe third Painleve equations of type $D_{6}$

as

follows

([28]):

$\{_{t\frac{}{dt}=-2qp^{2}+2pq-v_{1}p+\frac{1}{2}(v_{1}+v_{2})}^{t\frac{dq}{d_{p}^{t}}=2q^{2}p-q^{2}+v_{1}q+t}’$ .

Umemura and Watanabe show that (30) has transcendental classical solutions

if and only if$v_{1}+v_{2}$

or

$v_{1}-v_{2}$ is

an even

integer. For example, if $v_{1}+v_{2}=0$,

we have classical solutions$p=0$ and

$t \frac{dq}{dt}=-q^{2}+v_{1}q+t$. (31)

We will introduce

anew

variable $u$ by

$q= \frac{v_{1}}{2}+t\frac{d}{dt}(\log u)$.

Then (31) turns to be the linear equation

$\frac{d^{2}u}{dt^{2}}+\frac{1}{t}\frac{du}{dt}-\frac{1}{t^{2}}(t+\frac{v_{1}^{2}}{4})u=0$,

which is equivalent to Bessel’s equation.

If$v_{1}+v_{2}$

or

$v_{1}-v_{2}$ is

an even

integer, there exist aBacklund transformation

on

(30),

we can

reduce to the

case

$v_{1}+v_{2}=0$. Therefore any transcendental

classical solutions

can

be represented by Bessel functions.

(11)

The Backlund transformation group of the third Painleve equations oftype $D_{6}$ is the affine Wyle

group

$\ovalbox{\tt\small REJECT} \mathrm{T}^{\prime^{\ovalbox{\tt\small REJECT}}}(A.)\mathrm{e}\mathrm{I}\ovalbox{\tt\small REJECT} \mathrm{I}^{\ovalbox{\tt\small REJECT}}(A.)$ and walls of this action is the set

{

$(v_{1},v_{2})\in \mathbb{C}|v_{1}+v_{2}$

or

$v_{1}-v_{2}$ is

an even integer}

.

For other Painleve equations (second, fourth, fifth and sixth), they have

tran-scendental classical solutions

on

the walls ofaction ofthe affine Wyle group.

The third Painleve equations of type $D_{7}$ has

one

parameter $a$ and the

Backlund transformation

group

is the affine Wyle group $W(A_{1})$. Hence we

may expect that the equation (5) has transcendental classical solutions

on

the

walls. But the this expectation is incorrect. We will show that

Theorem 4.1. The third Painlevi equations

_of

type $D_{7}$ do not have

transcen-dental classical solutions.

In [2] algebraic solutions of the third Painleve equations of type $D_{7}$

are

classified.

Combined

with Theorem 4.1,

we

have

Theorem 4.2. The third Painlevi equations

_of

type $D_{7}$ have classical solutions

if

and only

_if

$a$ is

an

integer.

If

as

follows:

(J) For any ordinary differential field extension $K/\mathbb{C}(t)$, there exists

no

prin-cipal ideal I of$K[f,g]$ _{such that O} $\subsetarrow I$ C, $K[f,g]$ and $X(a)I\subset I$

.

Thefollowing Proposition is the key in this paper.

Proposition 4.3. The derivation $X(a)$ does not satisfy the condition (J).

Proof.

Assume

.

Let $R_{d}$ be the $K$-linear subspace of

$K[f,g]$ generated

over

$K$ by all the monomials of weight $d$

.

_{$R_{-d}=K[f^{2}g]f^{d}$},

$R_{2d}=K[f^{2}g]g^{d}$, $R_{2d-1}=K[f^{2}g]fg^{d}$, for any non-negative integer $d$ and

$K[f,g]=\oplus R_{d}d\in \mathrm{Z}$’ $R_{d}\cdot R_{d’}=R_{d+d’}\backslash$

.

We define three homogeneous derivations $X_{-2}$,$X_{0}$,$X_{1}$ by

$X_{1}=(2f^{2}g+t) \frac{\partial}{\partial f}-2fg^{2}\frac{\partial}{\partial g}$,

$x_{-2}x_{0}$ $==$ $t \frac{\partial}{\partial}-a.f\frac{\partial}{\partial f}+ag\frac{\partial}{\partial g}-\frac{1t}{2}\frac{\partial}{\partial g}$

,

We have $X(a)=X_{-2}+X_{0}+X_{1}$ _{and each} $X_{i}$ maps $R_{d}$ to _{$R_{d+:}$}.

In the second grading,

we

set the weights of $f$ and $g$ to be 2and -1

re-spectively. The weight of amonomial $af^{:}g^{j}$ in $K[f,g]$ is $2i-j$ for any _{$a\in K$}

$(a\neq 0)$

.

Let $R_{d}’$ be the$K$-linear subspace of$K[f,g]$ generated

over

_$K$ by all the

monomials ofweight $d$. $R_{-d}’=K[fg^{2}]g^{d}$, $R_{2d}’=K[fg^{2}]f^{d}$, $R_{2d-1}’=K[fg^{2}]f^{d}g$,

for any non-negative integer $d$ and

$K[f,g]=\oplus R_{d}’d\in \mathrm{Z}$’ $R_{d}’\cdot R_{d}’,$ $=R_{d+d’}’$.

We define three homogeneous derivations $X_{-2}’,X_{0}’$,$X\mathrm{i}$ by

$X_{1}’$ $=$ $2f^{2}g \frac{\partial}{\partial f}-(\frac{1}{2}+2fg^{2})\frac{\partial}{\partial g}$, $X_{0}’$ $=$ $t \frac{\partial}{\partial t}-af\frac{\partial}{\partial f}+ag\frac{\partial}{\partial g}$,

$X_{-2}’$ $=$ $t \frac{\partial}{\partial f}$.

We have $X(a)=X_{-2}’+X_{0}’+X_{1}’$ and each $X_{i}’$ maps $R_{d}’$ to $R_{d+:}’$.

(13)

The both gradings

come

from the Newton polygon of $X(a)$. The Newton

polygon of$X(a)$ is

as

follows:

$f$

Here

an

integral point $(i,j)$ representsthe derivation$bf^{:+1}g^{j}\partial/\partial p+cf^{:}g^{j+1}\partial/\partial g$

$(b,c\in K)$

.

Since

the Newton polygon of type $D_{7}$ is different from the polygon oftype $D_{6}$ in [28],

we

choose different gradings.

We will determine the polynimial $G$

.

Since the highest part $X_{1}$ and $X_{1}’$

are

of weight one, $G$ is at most ofweight

one

in the both degree. Therefore

$G=\lambda fg+\mu$

for

some

$\lambda,\mu\in K$

.

Step $\ell$

.

the highest part of F with respect to the first grading

We will consider the highest part of $F$

.

Let $F$ be

asum

of homobeneous

polynomials with respect tothe first grading

$F=F_{m’}’+F_{m’+1}+\cdots+F_{1}+\cdots+F_{m}$

.

$(m’\leq m)$

$F_{j}\in R_{j}$, Frpl,$F_{m}\neq 0$ and if$m=m’=0$, $F_{0}\not\in K$

.

The homogeneous part of

(33) is

$X_{1}F_{k-1}+X_{0}F_{k}+X_{-2}F_{k+2}=\lambda fgF_{k-1}+\mu F_{k}$

.

(33)

Firstly

we

claim that $F_{m}$ is not divisible by $f$

.

If$F_{m}$ is divisible by $f$, there

exists

an

integer $k\geq 1$

$F_{m}=Qf^{k}$, $f\{Q$, $Q\in K[f,g]$

.

invariant polynomial, $b^{-1}F$ is also

an

invariant polynomial

for any $b\in K$

.

Hence

we

may

assume

$b=1$. Prom

now

on,

we

assume

$b=1$ _in

(35).

Step 3. $\mathrm{F}2\mathrm{p}-\mathrm{i}$ and $\mathrm{F}2\mathrm{p}-2$

We will determine$F_{2p-1}$ and $\mathrm{F}2\mathrm{p}-2$. By (34), $F_{2p-1}\mathrm{F}2\mathrm{p}-2$ and $\mathrm{F}2\mathrm{p}_{-}3$ satisfy

the equations $X_{1}(F_{2p-1})+X_{0}(F_{2p})$ $X_{1}(F_{2p-2})+X_{0}(F_{2p-1})$ We

can

set $=$ $\lambda fgF_{2p-1}+\mu F_{2p}$, (36) $=$ $\lambda fgF_{2p-2}+\mu F_{2p-1}$. (37) $F_{2p-1}$ $=$ $g^{p}f \sum_{j=0}^{k_{1}}d_{j}L^{j}$, $F_{2p-2}$ $=$ $g^{p-1} \sum_{j=0}^{k_{2}}e_{j}L^{j}$,

for $d_{j}$,$e_{j}\in K$ and $L=f^{2}g+t$. By (35)

we

have

$\mu F_{2p}$ -Xl$( \mathrm{F}2\mathrm{p})=(\mu+\frac{a\lambda}{2})L^{p+\frac{\lambda}{2}}g^{p}-(a+1)t(p+\frac{\lambda}{2})L^{p+\frac{\lambda}{2}-1}g^{p}$. (38)

Moreover

we

have

$X_{1}(F_{2p-1})-\lambda fgF_{2p-1}$

$=g^{p} \sum_{j=0}^{k}d_{j}[(2-\lambda-2p+2j)L^{j+1}+(\lambda-1+2p-2j)tL^{j}]$ .

(39)

Comparing (38) and (39),

we

have$k=p+\lambda/2-1$. But in this

case

thecoefficient

of $L^{p+\lambda/2-1}g^{p}$ of $X_{1}(F_{2p-1})-\mathrm{X}\mathrm{f}\mathrm{g}\mathrm{F}2\mathrm{p}-\mathrm{i}$ becomes

zero.

Hence

we

have

$\mu+\frac{a\lambda}{2}=0$. (40)

Moreover

we

have

$4=0$, $d_{1}=0$,$\ldots$,$d_{k-1}=0$,$d_{k}=-(a+1)(p+ \frac{\lambda}{2})$ .

(15)

Finally

we

have

$F_{2p-1}=- \frac{1}{2}(a+1)(2p+\lambda)g^{p}f(f^{2}g+t)^{p+\lambda/2-1}$

.

In the

same

way

we

obtain

$F_{2p-2}=- \frac{1}{8}(a+1)^{2}(2p+\lambda-2)(2p+\lambda)tg^{p-1}(f^{2}g+t)^{p+\lambda/2-2}$,

from (37).

Step

_4.

The highest part of$F$ with respect to the second grading

Let

us

decompose $F$with respect to the second grading

$F=F_{n}’,$ $+F_{n+1}’,+\cdots+F_{1}’+\cdots+F_{n}’$

.

$(n’\leq n)$

$F_{j}’\in R_{j}’$, $F_{n}’,$,$F_{n}’\neq 0$ and if $n=n’=0$, $F_{0}’\not\in K$.

In the

same

way $F_{n}’$ is not divisible by _$g$ and $n$ is

an even

integer $2q$. There

exists anon-negative integer $k$ such that A $=2(q-k)$ and

$F_{2q}’=c( \frac{1}{2}+fg^{2})^{q-\lambda/2}f^{q}$ (41)

for $c\in K$

.

The Newton polygon of$F$ has the following figure:

5.

$F_{2q-1}’$ and $F_{2q-2}’$

We

can

calculate $F_{2q-1}’$ and $F_{2q-2}’$ in the

same

way

as

the Step 3:

$F_{2q-1}’= \frac{1}{2}a(\lambda-2q)f^{q}g(fg^{2}+\frac{1}{2})^{q-\lambda/2-1}$ ,

(16)

$F_{2q-2}’=- \frac{1}{16}a^{2}(\lambda-2q)(\lambda-2q+2)f^{q-1}(fg^{2}+\frac{1}{2})^{q-\lambda/2-2}$

We will compare $\mathrm{F}2\mathrm{P}-\mathrm{i}$ and _{$F_{2q-1}’$}. From (42)

we

have

$f^{2p+\lambda-1}g^{2p+\lambda/2-1}=f^{2q-\lambda/2-1}g^{2q-\lambda-1}$ .

The coeffiecient of this monomial in $F$ is

$- \frac{1}{2}(a+1)(2p+\lambda)=\frac{1}{2}a(\lambda-2q)$

.

From (42)

we

have

$\lambda=-\frac{4p}{a+2}$. (43)

Step 6. $F_{2p-3}’$

$F_{2p-3}’$ is determined by the equation

$X_{1}(F_{2p-3}’)+X_{0}(F_{2p-2}’)+X_{-2}(F_{2p}’)= \lambda fgF_{2p-3}’-\frac{a\lambda}{2}F_{2p-2}$’

We may

assume

that $F_{2p-3}’$ has the form

$F_{2p-3}’=f^{p-1}g \sum_{j=0}^{k_{3}}h_{j}M^{j}$,

where $M=fg^{2}+1/2$

.

Then

we

have

$\mu F_{2p-2}’-X_{0}(F_{2p-2}’)-X_{-2}(F_{2p}’)=\sum_{j=0}^{3}s_{j}f^{q-1}M^{q-\lambda/2-j}$,

where

$s_{0}=(- \frac{1}{2}\lambda+2q)t$, $s_{1}=( \frac{1}{4}\lambda-\frac{1}{2}q)t$, $s_{2}=-al$, $s_{3}=al(1+ \frac{\lambda}{4}-\frac{q}{2})$ ,

and

$l=- \frac{1}{16}a^{2}(\lambda-2q)(\lambda-2q+2)$.

For any positive integer $s$

we

have

$X_{1}(g^{p-1}M^{q-\lambda/2-\epsilon})-\lambda fg(g^{p-1}M^{q-\lambda/2-\epsilon})$

$=(2s-4)f^{q-1}M^{q-\lambda/2-s+1}+( \frac{3}{2}-s)f^{q-1}M^{q-\lambda/2-s}$.

(44) Therefore $k_{3}=q-\lambda/2-1$

.

Setting $s=1$ in (44),

we

have

$X_{1}(g^{p-1}M^{q-\lambda/2-1})-\lambda fg(g^{p-1}M^{q-\lambda/2-1})$

$=-2f^{q-1}M^{q-\lambda/2}+ \frac{1}{2}f^{q-1}M^{q-\lambda/2-1}$ .

(17)

Therefore

$h_{q-\lambda/2-1}=- \frac{s_{0}}{2}$ (45)

Setting $s=2$ in (45),

we

have

$X_{1}(g^{p-1}M^{q-\lambda/2-2})- \lambda fg(g^{p-1}M^{q-\lambda/2-2})=-\frac{1}{2}f^{q-1}M^{q-\lambda/2-2}$

and the term $f^{q-1}M^{q-\lambda/2-1}$ does not appear. Therefore

$\frac{1}{2}h_{q-\lambda/2-1}=s_{1}$. (46)

Comparing (45) and (46),

we

have

’this

is contradiction. $\square$

4.3 Proof of

Theorem

4.1

The derivation $X(a)$ satisfies the condition _(J) for any $a$

.

By Theorem 1.1 in

[27]

we see

that every transcendental solution of the equation of type $D_{7}$ is non-classical.

By aquadratic transformation, the third Painleve equation oftype $D_{8}$

re-duces to athird Painlev\’e equation oftype $D_{6}$

.

The third Painlev\’e equation of

type $D_{8}$

$y’= \frac{1}{y}y^{\prime 2}-\frac{y’}{x}+\frac{\alpha y^{2}+\beta}{x}$

has two algebraic solutions $y=\sqrt{-\beta}/\alpha$ and

no

transcendental classical

solu-tions.

Thus

we

classified classical solutions of the third Painleve equation of all

type.

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